[seqfan] Re: Primes adding one of their digit to themselves (+chains)

[Eric Angelini]

Hello SeqFans,
Paolo Lava has computed 1000 terms of S - many thanks to him
(I hope that the seq will be accepted in the OEIS if of interest)

[Reminder: all terms are primes; if you add to the term a(n)
one of its digits, you'll get another prime; example 29+2=31
or 43+4=47 or 61+6=67]

S = 29, 43, 61, 67, 89, 167, 227, 239, 263, 269, 281, 349, 367, 389, 439, 457, 461, 463, 487, 499, 521, 563, 601, 607, 613, 641, 643, 647, 653, 677, 683, 821, 827, 983, 1063, 1229, 1277, 1283, 1289, 1361, 1367, 1423, 1427, 1429, 1447, 1481, 1483, 1489, 1549, 1601, 1607, 1613, 1621, 1657, 1663, 1693, 1721, 1823, 1861, 1867, 1871, 2027, 2063, 2081, 2087, 2111, 2129, 2141, 2237, 2267, 2309, 2339, 2347, 2381, 2437, 2467, 2473, 2549, 2591, 2657, 2671, 2677, 2683, 2687, 2693, 2711, 2729, 2749, 2789, 2801, 2843, 2879, 2963, 2969, 2999, 3061, 3163, 3251, 3257, 3299, 3329, 3343, 3457, 3461, 3463, 3527, 3607, 3617, 3631, 3637, 3671, 3691, 3761, 3821, 3847, 3881, 3929, 3943, 4003, 4127, 4129, 4153, 4217, 4229, 4241, 4259, 4271, 4289, 4421, 4447, 4513, 4519, 4561, 4583, 4637, 4639, 4643, 4651, 4657, 4673, 4721, 4729, 4783, 4789, 4813, 4933, 4967, 4969, 4999, 5021, 5189, 5231, 5279, 5347, 5413, 5437, 5479, 5563, 5641, 5647, 5651, 5653, 5683, 5783, 5813, 5843, 5849, 5861, 6037, 6043, 6047, 6067, 6073, 6197, 6211, 6257, 6263, 6269, 6271, 6299, 6311, 6317, 6323, 6337, 6353, 6361, 6367, 6373, 6389, 6421, 6469, 6547, 6563, 6571, 6653, 6673, 6703, 6823, 6827, 6833, 6857, 6863, 6899, 6911, 6961, 6971, 6977, 6983, 6991, 7127, 7211, 7243, 7477, 7481, 7583, 7643, 7681, 7867, ...
The "chains" have also a certain charm, no? Paolo computed a few
of those - and I think that the first term of each line could also
produce a nice seq.

[Reminder: to jump from one prime 'p' to the next one in the "chain",
add to 'p' one of its digits]
PL> first chains of 4 terms:

   601 -> 607 -> 613 -> 619

PL> first chains of 5 terms:
    6257 -> 6263 -> 6269 -> 6271 -> 6277
PL> first chains of 6 terms:
    19417 -> 19421 -> 19423 -> 19427 -> 19429 -> 19433
PL> and so on... Here the first chain with 10 primes:
    246899 -> 246907 -> 246913 -> 246919 -> 246923 -> 246929 -> 246931 -> 246937 -> 246941 -> 246947
Bravo, Paolo!

------------
Now, what about alternating primes 'p' and non-primes 'q'?

Starting with a 'p', we have this 10-element alternating "chain",
for instance:


T = 5-10-11-12-13-16-17-18-19-28 END

    p  q  p  q  p  q  p  q  p  q



As always, the difference 'd' between two terms a(n) and a(n+1)

must be one of the digits of a(n).



Best,

É.
------------



P.-S.

Infinite chains of non-primes having this property are easy

to find; here is one:

U = 2,4,8,16,22,24,28,36,42,44,48,56,62,64,68,76,82,...

[always add to a(n) its last digit]